**Cartesian equation: **

((1 - *m*^{2})(*x*^{2} + *y*^{2}) + 2*m*^{2}*cx* + *a*^{2} - *m*^{2}*c*^{2})^{2} = 4*a*^{2}(*x*^{2} + *y*^{2})

This curve

The curves were first studied by Descartes in 1637 and are sometimes called the 'Ovals of Descartes'.

The curve was also studied by Newton in his classification of cubic curves.

The Cartesian Oval has bipolar equation *r* + *mr*' =*a*.

If *m* = ±1 then the Cartesian Oval *C* is a central conic while if *m* = *a*/*c* then the curve is a Limacon of Pascal (Étienne Pascal). In this case the inside oval touches the outside one.

Cartesian Ovals are anallagmatic curves.

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JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Cartesian.html